In a pot, combine beef brisket, water, onion, green onion, and garlic cloves. Cook over medium-low heat for about 1.5 hours. Skim off the foam. Discard the vegetables, keep the stock, and set aside the meat from the stock. Shred the meat using a knife or hands and add sesame oil to the meat.
Soak the rice cakes in cold water for 15 minutes. Discard the water after 15 minutes. Meanwhile, prepare the toppings.
Boil the dumplings in a pot. Set aside.
Boil the stock in a clean pot. Add minced garlic, soy sauce, rice cakes, and dumplings. Cook for 2-3 minutes. Adjust the taste by adding salt and add a pinch of black pepper. Serve the soup in a bowl and add toppings.